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Mirrors > Home > MPE Home > Th. List > Mathboxes > deccarry | Structured version Visualization version GIF version |
Description: Add 1 to a 2 digit number with carry. This is a special case of decsucc 11550, but in closed form. As observed by ML, this theorem allows for carrying the 1 down multiple decimal constructors, so we can carry the 1 multiple times down a multi-digit number, e.g. by applying this theorem three times we get (;;999 + 1) = ;;;1000. (Contributed by AV, 4-Aug-2020.) (Revised by ML, 8-Aug-2020.) (Proof shortened by AV, 10-Sep-2021.) |
Ref | Expression |
---|---|
deccarry | ⊢ (𝐴 ∈ ℕ → (;𝐴9 + 1) = ;(𝐴 + 1)0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dec 11494 | . 2 ⊢ ;(𝐴 + 1)0 = (((9 + 1) · (𝐴 + 1)) + 0) | |
2 | 9nn 11192 | . . . . . . . 8 ⊢ 9 ∈ ℕ | |
3 | peano2nn 11032 | . . . . . . . 8 ⊢ (9 ∈ ℕ → (9 + 1) ∈ ℕ) | |
4 | 2, 3 | ax-mp 5 | . . . . . . 7 ⊢ (9 + 1) ∈ ℕ |
5 | 4 | a1i 11 | . . . . . 6 ⊢ (𝐴 ∈ ℕ → (9 + 1) ∈ ℕ) |
6 | peano2nn 11032 | . . . . . 6 ⊢ (𝐴 ∈ ℕ → (𝐴 + 1) ∈ ℕ) | |
7 | 5, 6 | nnmulcld 11068 | . . . . 5 ⊢ (𝐴 ∈ ℕ → ((9 + 1) · (𝐴 + 1)) ∈ ℕ) |
8 | 7 | nncnd 11036 | . . . 4 ⊢ (𝐴 ∈ ℕ → ((9 + 1) · (𝐴 + 1)) ∈ ℂ) |
9 | 8 | addid1d 10236 | . . 3 ⊢ (𝐴 ∈ ℕ → (((9 + 1) · (𝐴 + 1)) + 0) = ((9 + 1) · (𝐴 + 1))) |
10 | 4 | nncni 11030 | . . . . . 6 ⊢ (9 + 1) ∈ ℂ |
11 | 10 | a1i 11 | . . . . 5 ⊢ (𝐴 ∈ ℕ → (9 + 1) ∈ ℂ) |
12 | nncn 11028 | . . . . 5 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℂ) | |
13 | 1cnd 10056 | . . . . 5 ⊢ (𝐴 ∈ ℕ → 1 ∈ ℂ) | |
14 | 11, 12, 13 | adddid 10064 | . . . 4 ⊢ (𝐴 ∈ ℕ → ((9 + 1) · (𝐴 + 1)) = (((9 + 1) · 𝐴) + ((9 + 1) · 1))) |
15 | 11 | mulid1d 10057 | . . . . . 6 ⊢ (𝐴 ∈ ℕ → ((9 + 1) · 1) = (9 + 1)) |
16 | 15 | oveq2d 6666 | . . . . 5 ⊢ (𝐴 ∈ ℕ → (((9 + 1) · 𝐴) + ((9 + 1) · 1)) = (((9 + 1) · 𝐴) + (9 + 1))) |
17 | df-dec 11494 | . . . . . . 7 ⊢ ;𝐴9 = (((9 + 1) · 𝐴) + 9) | |
18 | 17 | oveq1i 6660 | . . . . . 6 ⊢ (;𝐴9 + 1) = ((((9 + 1) · 𝐴) + 9) + 1) |
19 | id 22 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℕ) | |
20 | 5, 19 | nnmulcld 11068 | . . . . . . . 8 ⊢ (𝐴 ∈ ℕ → ((9 + 1) · 𝐴) ∈ ℕ) |
21 | 20 | nncnd 11036 | . . . . . . 7 ⊢ (𝐴 ∈ ℕ → ((9 + 1) · 𝐴) ∈ ℂ) |
22 | 2 | nncni 11030 | . . . . . . . 8 ⊢ 9 ∈ ℂ |
23 | 22 | a1i 11 | . . . . . . 7 ⊢ (𝐴 ∈ ℕ → 9 ∈ ℂ) |
24 | 21, 23, 13 | addassd 10062 | . . . . . 6 ⊢ (𝐴 ∈ ℕ → ((((9 + 1) · 𝐴) + 9) + 1) = (((9 + 1) · 𝐴) + (9 + 1))) |
25 | 18, 24 | syl5req 2669 | . . . . 5 ⊢ (𝐴 ∈ ℕ → (((9 + 1) · 𝐴) + (9 + 1)) = (;𝐴9 + 1)) |
26 | 16, 25 | eqtrd 2656 | . . . 4 ⊢ (𝐴 ∈ ℕ → (((9 + 1) · 𝐴) + ((9 + 1) · 1)) = (;𝐴9 + 1)) |
27 | 14, 26 | eqtrd 2656 | . . 3 ⊢ (𝐴 ∈ ℕ → ((9 + 1) · (𝐴 + 1)) = (;𝐴9 + 1)) |
28 | 9, 27 | eqtrd 2656 | . 2 ⊢ (𝐴 ∈ ℕ → (((9 + 1) · (𝐴 + 1)) + 0) = (;𝐴9 + 1)) |
29 | 1, 28 | syl5req 2669 | 1 ⊢ (𝐴 ∈ ℕ → (;𝐴9 + 1) = ;(𝐴 + 1)0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 (class class class)co 6650 ℂcc 9934 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 ℕcn 11020 9c9 11077 ;cdc 11493 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-ltxr 10079 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-dec 11494 |
This theorem is referenced by: (None) |
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